docs/PROOF_AUDIT.md

# Internal proof and implementation audit

This is an internal audit, not external peer review or proof-assistant verification.

## Main failure modes addressed

**Dependence across a cut.** The scalar iid inequality fails for a stationary two-state binary Markov source: `c2(1)=17/25`, `c2(2)=11/25`, and `c2(1)^2-c2(2)=14/625`. The release retains both hidden boundaries and applies the power inequality to the exact Markov convolution. The counterexample targets that independence extrapolation, not the Salem equality.

**Remainders and periodicity.** Proving a bound only on multiples of a block length would not suffice. The proof uses `N=kn+r`, positive row sums of every remainder matrix, and the growth of the sum of nonnegative matrix powers. Reducible and periodic matrices are included.

**The Shannon limit.** The apparent division by `1-q` does not justify simply substituting `q=1`. The proof differentiates the finite-block Perron eigenvalue, obtains the conditional entropy `H(X_n,Z_n|Z_0)`, and uses the order-independent error. For a periodic irreducible source, take a subsequence of block lengths coprime to its period.

**Nonergodic mixtures.** Different components can have different Shannon rates. The release explicitly proves the max/mean/min selection law and includes a discontinuity example. It does not force continuity by silently imposing irreducibility.

**Full maps versus numerators.** Unequal clocks make a numerator-only state insufficient. The code records completed-branch counters, synchronization time, and exact numerator differences. Finished replicas are never artificially padded.

**Probability at edge selection.** In the clock-slot expansion, edge probability is paid once, when the branch starts. Busy countdown steps are deterministic. This prevents accidentally raising an edge probability to its clock length.

**Safe pruning.** Gates are outward rational embedding intervals. Terminal carries are compared as exact integer coefficient vectors. Adaptive inverse-norm absorption bounds keep interval widening within a constant-thickness enlargement of the polynomial cone.

**Perron roots without unproved floating decisions.** Exact rational resolvent tests enclose the spectral radius and handle reducibility, periodicity and singular midpoints. A singular midpoint is assigned to the lower side only because no singularity can occur above the spectral radius.

**Complexity quantifiers.** A fixed field alone is not enough to promise uniform polynomial complexity in arbitrary digit bit length using geometric width bounds. The manuscript explicitly fixes the finite allowed digit and clock alphabets as part of the geometry. Noninteger orders and a generic Jordan compiler are not claimed as executable features.

**Dimension overreach.** Only the stated unit-clock grid dimensions are derived. A nonergodic mixture need not have one common pointwise dimension, and unequal clocks do not justify dividing a general Rényi rate by an average clock.

**Equality versus approximation.** Convergent certified entropy brackets do not identify their limit with `log(beta)`. No finite test, fitted trend, or continuity-at-one argument is promoted to a Salem equality proof.

## Development corrections made before release

An early conditional-mass routine reused the name of its optional terminal-state parameter as a loop variable. Exact reference comparisons exposed the error; the local variable was renamed, and the final tests pass. The spectral bracket was also tightened by including the maximum diagonal entry in its lower bound, allowing exact recognition of the reducible mixture's unit spectral radius. These are recorded corrections, not unresolved defects known to be present in the final implementation.

## Residual uncertainty

Finite tests do not rule out every implementation bug. The proof text has not been mechanically formalized or checked by another mathematician. The neutral-Jordan extension is not tested through a generic matrix implementation. The literature search is targeted rather than exhaustive. These limitations remain visible in the README, manuscript, status file and claim ledger.